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Forum:SOA's ordinal
Is it correct that fundamental sequence for SOA's ordinal supposed to be \alpha0 = 1 and C(\alphan+1,0) = C(\Omega^{\alphan},0) in Taranovsky's notation? If so, I guess that natural continuation of \psi(\Omega), \psi(M), \psi(K), \cdots indeed might reach SOA (alternative guess is C(\Omega^{\Omega^\omega},0) ). Also, how \alpha -order logic compares to \alpha -order arithmetic? Ikosarakt1 (talk ^ ) 11:56, June 16, 2014 (UTC) : First, I believe correct FS would be \(\alphan=C(\Omega_n,0)\). Second, I don't really think there is any "natural continuation" of \(\Omega,I,M,K\). One could then use indescribable cardinals, but there is large hierarchy of these, so I don't know how one would go about connecting these into sequence of length \(\omega\). : About the other question, I recently came to a conclusion that speaking about ordinals of n-th order logic doesn't really make sense, because logics don't prove anything - being n-th order is just a property of system we are working with. So, for example, while SOA can prove well-foundedness of \(\omega\), asking if SOL can prove this is quite meaningless. Grasping all of this is still quite a challenge for me, though. LittlePeng9 (talk) 13:05, June 16, 2014 (UTC) :Wait, if this notation can be extended for \Omega_n , then why not extend it up to \Omega_\alpha for arbitrary \alpha , I, M and K? By the way, there might be the ordinal so that C(\alpha,0) = \psi(\alpha) (catching ordinal) and we can define catching function for this (through probably it won't work, FS's for all intermediate ordinals have to be defined.) Ikosarakt1 (talk ^ ) 14:25, June 16, 2014 (UTC) ::This "catching function" sounds nice. Have y'all's come up with any formal proofs demonstrating its strength? you're.so. 14:49, June 16, 2014 (UTC) ::First, can we define a single, natural FS for arbitrary recursive ordinal? If yes, then catching function is at least total and would be defined for all \alpha . Look at SGH-catching-FGH function. If we'd define FS's only for ordinal up to BHO, it would be undefined at all as there are no catches. Ikosarakt1 (talk ^ ) 15:16, June 16, 2014 (UTC) ::Depends on what you mean by "natural"... LittlePeng9 (talk) 16:33, June 16, 2014 (UTC) :::Due to the very nature of recursive ordinals, it doesn't seem that your "natural" condition can be satisfied. However, Kleene's O can give us FS's. Suppose \(\alpha = \mathcal{O}(3 \cdot 5^i)\) for minimal \(i\); then \(\alphak = \mathcal{O}(f_i(k))\). you're.so. 05:56, June 17, 2014 (UTC) :::What this definition says for FS of, say, \psi(\psi_{\Omega_{\omega+1}}(0)) ? Is it possible to determine it for any reasonable time? Ikosarakt1 (talk ^ ) 06:01, June 17, 2014 (UTC) ::::@Iko No, given constraint that i is minimal this is uncomputable. LittlePeng9 (talk) 06:11, June 17, 2014 (UTC) :::::Peng is right; although theoretically it may be possible to find a working value of i and to prove that it is optimal, it is very difficult in the same sense as trying to compute \(\Sigma(10)\) (say). At this level of mathematics, we cannot rely on explicit examples because these entities are by nature inaccessible to us. you're.so. 07:07, June 17, 2014 (UTC) :::::But it isn't good because we won't know the power of catching function. Can we define it easier-to-determine by human mind (but still uncomputable)? Ikosarakt1 (talk ^ ) 07:45, June 17, 2014 (UTC) ::::::Don't know the power of the catching function? Then find it. Figure it out. Using proofs! 'Cause that's what you do at this point. ::::::Let me clarify further that you've reached a fundamental restriction on our physical universe by the Church-Turing thesis. We can't work by intuition and examples anymore. you're.so. 14:07, June 17, 2014 (UTC) :::::::Oops. For some reason this sounded a lot meaner than I intended it to be. Sorry about that. you're.so. 15:32, June 17, 2014 (UTC) ::::::It's the same as when we ask "Can we define easier/better/whatever Busy Beaver function?" The answer is - we can, but there are limitations. For example, we can create a made-up function by setting \(F(n)=n+1\) for \(n<10^6\) and \(F(n)=BB(n)\) for \(n\geq 10^6\). Result? We get function easy to determine for quite a bit of n's, but still catching up with Sigma function. But it's weird to define such functions. "Natural" examples of uncomputable functions take unpredictable values quickly. LittlePeng9 (talk) 14:50, June 17, 2014 (UTC) extending the notation past \(\Omega_n\) to transfinite \(\Omega_\alpha\) would be the obvious extension to make. The notation for \(\Omega_n\) depends on the term "n-built from below". To me, the obvious definition of "c-built from below" for arbitrary ordinals c would be: 1. a is 0-built from below from b if a <= b. 2. a is c+1-built from below from b iff the standard representation of a does not use ordinals above a except in the scope of an ordinal c-built from below from b. 3. if c is a limit ordinal, a is c-built from below from b iff a is d-built from below from b for some d < c. I don't know if this works or not... perhaps the notation is no longer well-founded once you go past \(\omega\). Deedlit11 (talk) 17:47, July 3, 2014 (UTC)